Once we have found the critical points, one way to determine if they are local minima or
maxima is to apply the first derivative test. Another way uses the second derivative of f.
Suppose x_{0} is a critical point of the function f (x), that is, f'(x_{0}) = 0. We
have the following three cases:

f''(x_{0}) > 0 implies x_{0} is a local minimum

f''(x_{0}) < 0 implies x_{0} is a local maximum

f''(x_{0}) = 0 is inconclusive

The first two of these options derive from the observation that f''(x_{0}) is the rate
of change of f'(x) at x_{0}, which will be positive if the derivative crosses zero
from negative to positive, and negative is the derivative crosses zero from positive to
negative. This is called the second derivative test for maxima and minima. The
third, inconclusive case is considered below.

The first and second derivative tests employ essentially the same logic, examining what
happens to the derivative f'(x) near a critical point x_{0}. The first derivative
test says that maxima and minima correspond to f' crossing zero from one direction or
the other, which is indicated by the sign of f' near x_{0}. The second derivative
test is just the observation that the same information is encoded in the slope of the
tangent line to f'(x) at x_{0}.

Concavity and Inflection Points

A function f (x) is called concave up at x_{0} if f''(x_{0}) > 0, and concave
down if f''(x_{0}) < 0. Graphically, this represents which way the graph of f is
"turning" near x_{0}. A function that is concave up at x_{0} lies above
its tangent line in a small interval around x_{0} (touching but not crossing at x_{0}).
Similarly, a function that is concave down at x_{0} lies below its
tangent line near x_{0}.

The remaining case is a point x_{0} where f''(x_{0}) = 0, which is called an inflection
point. At such a point the function f holds closer to its tangent line than
elsewhere, since the second derivative represents the rate at which the function turns
away from the tangent line. Put another way, a function usually has the same value and
derivative as its tangent line at the point of tangency; at an inflection point, the
second derivatives of the function and its tangent line also agree. Of course, the
second derivative of the tangent line function is always zero, so this statement is
just that f''(x_{0}) = 0.

Inflection points are the critical points of the first derivative f'(x). At an
inflection point, a function may change from being concave up to concave down (or the
other way around), or momentarily "straighten out" while having the same concavity to
either side. These three cases correspond, respectively, to the inflection point x_{0}
being a local maximum or local minimum of f'(x), or neither.